Improved partial permutation decoding for Reed-Muller codes

It is shown that for n ź 5 and r ź n - 1 2 , if an ( n , M , 2 r + 1 ) binary code exists, then the r th-order Reed-Muller code R ( r , n ) has s -PD-sets of the minimum size s + 1 for 1 ź s ź M - 1 , and these PD-sets correspond to sets of translations of the vector space F 2 n . In addition, for the first order Reed-Muller code R ( 1 , n ) , s -PD-sets of size s + 1 are constructed for s up to the bound ź 2 n n + 1 ź - 1 . The results apply also to generalized Reed-Muller codes.

[1]  Washiela Fish,et al.  Partial permutation decoding for simplex codes , 2012, Adv. Math. Commun..

[2]  Jennifer D. Key,et al.  Information sets and partial permutation decoding for codes from finite geometries , 2006, Finite Fields Their Appl..

[3]  Mohammed S. EL-Atrash,et al.  Linear Codes over Finite Fields , 2018, Designs from Linear Codes.

[4]  Jennifer D. Key,et al.  Partial permutation decoding for codes from finite planes , 2005, Eur. J. Comb..

[5]  Rita Procesi,et al.  Codes and groups , 2006 .

[6]  Daniel M. Gordon.  Minimal permutation sets for decoding the binary Golay codes , 1982, IEEE Trans. Inf. Theory.

[7]  J. Macwilliams.  Permutation decoding of systematic codes , 1964 .

[8]  Eugene Prange,et al.  The use of information sets in decoding cyclic codes , 1962, IRE Trans. Inf. Theory.

[9]  Mercè Villanueva,et al.  Partial permutation decoding for binary linear Hadamard codes , 2014, Electron. Notes Discret. Math..

[10]  Richard A. Brualdi,et al.  Greedy Codes , 1993, J. Comb. Theory, Ser. A.

[11]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[12]  Jennifer D. Key,et al.  Reed-Muller codes and permutation decoding , 2010, Discret. Math..

[13]  Ray Hill,et al.  A First Course in Coding Theory , 1988 .

[14]  Hans-Joachim Kroll,et al.  PD-sets for the codes related to some classical varieties , 2005, Discret. Math..

[15]  Jennifer D. Key,et al.  Designs and their codes , 1992, Cambridge tracts in mathematics.

[16]  E. F. Assmus.  POLYNOMIAL CODES AND FINITE GEOMETRIES , 2003 .